In a group of n women and n men, how many people must be selected at random from the total 2n population to guarantee that a woman and man couple is selected?

A product shipment contains 6 computer chips, of which two (2) are defective. If an inspector selects 4 chips at random for an order, without replacement, how many ways are there of picking both defective chips in this order (of 4 chips)?  Order does not matter.

Consider a deck of 52 cards with 4 suits and 13 cards (2-10,J,K,Q,A) in each suit. Jack takes one such deck and arranges them in a line in a completely random order. Now he wants to find the number of "Power Trios" in this line of cards. A "Power Trio" is a set of 3 consecutive cards where all cards are either a Jack, Queen or King (J,Q or K). A "Perfect Power Trio" is a set of 3 consecutive cards with exactly 1 Jack, 1 Queen and 1 King (in any order). Find the expected number of Power Trios that Jack will find. Find the expected number of Perfect Power Trios that Jack will find.

A standard deck of cards consists of 52 cards n each of the 4 suits: Spades, Diamonds, Hearts and Clubs. Each suit contains 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen and King. A “hand" is a selection of 5 cards from the deck. Thus there are (5²) possible hands. Suppose three players are each dealt a hand of 5 cards each from a shuffled deck. What is the chance that the first player gets a royal flush (i.e 10-J-Q-K-A of the same suit)? What is the chance that at least one player obtains a royal flush?

A bag contains pieces of 6 different types of candy. If the bag has 85 pieces in total, how many pieces must be randomly selected to guarantee the selection contains at least 3 pieces of one of the 6 types?

In a fuel economy study, each of 3 race cars is tested using 5 different brands of gasoline at 7 test sites located in different regions of the country. If 2 drivers are used in the study, and test runs are made once under each distinct set of conditions, how many test runs are needed?

If two states are selected at random from a group of 10 states, determine the number of possible outcomes if the group of states are selected with replacement or without replacement. If the states are selected with replacement, there are possible outcomes.

1. Assume that a class of 16 students is supposed to write an essay. If 3 essays are randomly selected to appear on the class bulletin board, how many different groups of 3are possible? (calculate number of combinations of 3 items from 16 available items).Show the result of calculations.

A teacher will randomly award three students with extra credit. How many possible ways can the teacher pick these three students from a class of 20 students?

A husband and wife, Stan and Lucretia, share a digital music player that has a feature that randomly selects which song to play. A total of 2643 songs have been loaded into the player, some by Stan and the rest by Lucretia. They are interested in determining whether they have loaded different proportions of songs into the player. Suppose that when the player was in the random-selection mode, 27 of the first 40 songs selected were songs loaded by Lucretia. Let p denote the proportion of songs that were loaded by Lucretia. a.) State the null and alternative hypotheses to be tested. How strong is the evidence that Stan and Lucretia have loaded different proportions of songs into the player? Make sure to check the conditions for the use of this test